example of four exponentials conjecture

Taking x1=iπ, x2=iπ2, y1=1, y2=2, we see that this conjecture implies that one of eiπ, eiπ2, or ei2π is transcendental. Since the first is -1 and the last is 1, the conjecture states that second must be transcendental, that is, eiπ2 is (conjecturally) transcendental.

In this particular case, the result is known already, so the conjecture is verified. Using Gelfond’s theorem, take α=eiπ and β=2 and it follows that αβ is transcendental.

Title example of four exponentials conjectureMathworldPlanetmath
Canonical name ExampleOfFourExponentialsConjecture
Date of creation 2013-03-22 14:09:09
Last modified on 2013-03-22 14:09:09
Owner archibal (4430)
Last modified by archibal (4430)
Numerical id 6
Author archibal (4430)
Entry type Example
Classification msc 11J81